Bring large pot of water to boil over high heat to cook pasta.
Clean portobello mushrooms: Remove and discard stems from each mushroom. Then, using spoon, scrape black gills from underside of each mushroom and discard gills. Cut each portobello mushroom in half; thinly slice each half.
In large nonstick pan over high heat, add portobello mushroom slices. Stirring constantly, cook until portobello mushrooms are softened, 4 to 5 minutes. Reduce heat to medium-high and add in sliced white mushrooms. Stirring occasionally, cook until white mushrooms are also soft, 10 to 12 minutes.
Meanwhile, add pasta to boiling water. Cook for amount of time package directions indicate for al dente. Reserve ½ cup pasta water for later use and drain pasta in colander.
When mushrooms are softened, reduce heat to low and stir in cream cheese, sour cream, 1 teaspoon fresh thyme, salt, and pepper. Stir and let cook 1 to 2 minutes until cream cheese is melted. Stir in reserved pasta water and remove from heat.
Add cooked pasta into the sauce and use tongs to combine. Divide into 4 portions, garnishing each portion with Parmesan cheese and remaining fresh thyme. Serve.